Trending questions
159,037 questions
3
votes
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answer
196
views
Surjectivity of specialization map
Let $S$ be a henselian DVR and $X/S$ be a flat and proper curve with $X$ being regular. Under what conditions the specialization map $Pic^0_{X/S}(S)\to Pic^0_{X/S}(Spec(k(s)))$ is surjective? Here $s\...
- 918
0
votes
2
answers
100
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Is there an uncountable extension of the Ramsey set $[\omega]^2$?
We say that a family ${\cal A}\subseteq {\cal P}(\omega)$ is Ramsey
if for every map $c:{\cal A}\to\{0,1\}$ there is an infinite set $X\subseteq \omega$
with the following properties:
${\cal A}\cap {\...
- 48.9k
5
votes
0
answers
87
views
Application of character sheaves to characters of $G(\mathbb{F}_q)$
I wish to ask about good examples of new applications of Lusztig's theory of character sheaves (and subsequent development, but excluding generalized Springer theory) back to the theory of characters ...
- 1,856
5
votes
2
answers
238
views
Residue of Dirichlet series at $s = 1$
Let $(a_n)_{n \ge 1}$ be a sequence of complex numbers, and suppose that the sequence has a well-defined "average", in the sense that
$$ \lim_{N \to \infty} \frac{1}{N}\sum_{i = 1}^N a_i = R$...
- 37k
0
votes
0
answers
140
views
State of the art on attempts to solve the elliptic curve discrete logarithm problem through transfering the problem to a weaker curve
Let an elliptic curve $E$, and 2 points on such curve $P$ and $O$ the methods I’m talking about consist in creating a weaker elliptic curve $F$ and mapping $P$ and $O$ to $F$ while successfully ...
- 87
4
votes
0
answers
82
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Classification of nilpotent orbits over local fields (for type ABCD via partitions )
Let $\mathfrak g$ be a simple Lie algebra over a char $0$ local field $F$ (e.g. $F=\mathbb R$ or $F=\mathbb Q_p$) with its adjoint group $G$. Let $\mathcal N \subseteq \mathfrak g$ be its nilpotent ...
- 6,622
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votes
0
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31
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Looking for a citation for this simple generalization of the Markov bound to non-negative super-martingales
Does anybody know a reference for the following theorem?
Theorem 1. Let $(X_t)_{t=0}^\infty$ be a non-negative supermartingale.
Then, for any constant $c > 0$, the event $(\exists
> t)\, X_t \...
- 183
0
votes
0
answers
79
views
Alternative proof of parabolic implosion
I am working on an alternative proof of parabolic implosion from complex dynamics, but only allowing hyperbolic perturbation.
Theorem (Parabolic Implosion) Let $f(z)=z^2+z$ and $U_f$ be parabolic ...
3
votes
0
answers
67
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$(M_1\otimes M_2)\otimes_{A\otimes A} A\cong M_1\otimes_A M_2$ in HA
I apologize for this naive question, I believe it's somewhere in HA but I just can't find it.
For symmetric monoidal category $C$, consider $A\in CAlg(C)$, and $M,N\in Mod_A(C)$, we can formulate the ...
- 618
4
votes
1
answer
175
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Maximizing trace subject to two equality constraints
I am looking at the following optimization problem
$$\begin{align}
\underset{{\bf X}}{\text{maximize}} \qquad&\mathrm{tr}({\bf AX})\\
\text{subject to} \qquad& \mathrm{tr}({\bf X}) = 1,\\
&...
- 43
4
votes
1
answer
375
views
Is the geometric realization of simplicial functors interesting?
While studying a completely unrelated problem, I have proved something on the following line: given a diagram of simplicial sets $X: C \to \textrm{sSet}$, some deformations of the geometric ...
- 2,152
51
votes
3
answers
5k
views
Is there mathematical significance to the LaGuardia floor tiles?
I noticed that the new terminal at LaGuardia Airport (in New York) has an intriguing design for the tiles on at least one of their floor areas. It bears a superficial resemblance to aperiodic tilings ...
- 677
2
votes
0
answers
68
views
Two-terms Euler-Maclaurin formula for concave functions over polytopes
Let $P\subset\mathbb{R}^{n}$ be a lattice polytope (vertices are
in $\mathbb{Z}^{n}$). Set $P_{k}=P\cap k^{-1}\mathbb{Z}^{n}$ for
$k\geq1$. Given a concave function $\phi:P\rightarrow\mathbb{R}$
(...
- 73
3
votes
1
answer
160
views
How to check whether a triangulated subcategory is admissible?
Let $\mathcal{T}' \subseteq \mathcal{T}$ be a full triangulated subcategory. Recall, $\mathcal{T}'$ is called $\textit{right admissible}$ if the inclusion $\mathcal{T}' \hookrightarrow \mathcal{T}$ ...
- 127
8
votes
1
answer
257
views
Maximal Ramsey families
We say that a family $\mathcal R\subseteq \mathcal P(\omega)$ is Ramsey if
$\bigcup \mathcal R = \omega$, and
for every map $f:\mathcal R \to \{0,1\}$ there is an infinite set $X\subseteq \omega$
...
- 48.9k
5
votes
1
answer
379
views
Factorisation of division polynomial
Let $\Psi_n$ denote the $n$-th division polynomial associated with the elliptic curve $y^2 = x^3 + A$, where $n$ is a natural number. The division polynomials are defined recursively, as described in ...
- 127
29
votes
4
answers
2k
views
What choice principles does "every set is in bijection with a transitive set" imply?
A set $X$ is transitive if $x\in y\in X\implies x\in X$. We shall say $\mathsf{TC}$ to mean the axiom "For all $X$ there is transitive $Y$ such that $|X|=|Y|$", where $|X|=|Y|$ means that ...
- 2,206
5
votes
0
answers
216
views
Are there Sudoku variants which are useful or mathematically deep?
I was recently watching a Sudoku Youtube channel which shows a large number of variants on the traditional Sudoku puzzle, some of them non-trivial to solve. I think there was some mention of a Sudoku ...
- 5,146
1
vote
0
answers
42
views
Why does the Kieboom characterization of shape is restricted only to paracompact spaces?
Borsuk founded shape theory as an extension of homotopy theory, appropriate for spaces with bad local properties. Borsuks definition was applied only to compact metric spaces. Later, this was ...
- 111
7
votes
2
answers
877
views
What is the best way to read advanced textbooks in Pure Mathematics (PhD Level)? [closed]
This question was asked earlier on Mathstackexchange but was closed very very soon without any answer and then deleted by the system!
I am a PhD student (1st year) in a poor country with a corrupt ...
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2
votes
1
answer
216
views
Compute the singular homology group modulo barycentric subdivision
Let $X$ be a topological space, and let $C(X)$ denote its singular chain complex with boundary operator $\partial$ and $n$-th chain group $C_n$. We know there exists a barycentric subdivision operator ...
- 807
1
vote
1
answer
180
views
Unitary representations of discrete (locally compact) groups
Let $\Gamma$ be a discrete (locally compact) subgroup of a locally compact Lie group. Let $H = L^2(\mathbb R^n, \mathbb C)$. Assume we have a complex unitary representation $$\Phi : \Gamma \to \...
- 357
2
votes
0
answers
98
views
An injective map in equivariant algebraic K-theory
Let $G$ be a simply connected simple algebraic group over $\mathbb{C}$ and $\mathfrak{g}$ be its Lie algebra. Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ consists of all nilpotent ...
- 653
0
votes
0
answers
113
views
Are measures singular with respect to all representing measures in $\mathbb{D}^n$ always concentrated on null-sets?
Let $\mu$ and $\nu$ be two measures on the $\sigma$-Borel set $\mathcal{B}(\mathbb{D}^n)$.
We say that $\mu$ is a representing measure for some point $z \in \mathbb{D}^n$, if
$$\forall_{u \in A(\...
- 63
1
vote
1
answer
80
views
What are the efficient algorithms to compute Hamiltonian paths on Cayley graphs of finite groups ? Can GAP do it?
The famous Lovasz conjecture predicts existence of the Hamiltonian path on Cayley graphs. In general finding such a path is NP-complete problem, but there are many heuristic algorithms.
Question 1: ...
- 24.9k
1
vote
0
answers
91
views
How to optimize parametric information-theoretic bounds?
I am faced with an information-theoretic upper bound, such as
\begin{align}
\sqrt{\alpha'}2^{I_\alpha(X;Y)},
\end{align}
where $I_\alpha(X;Y)$ is the Rényi mutual information with parameter $\alpha>...
- 287
-4
votes
0
answers
135
views
Which arithmetic\set theory is synonymous with this theory?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x$
Define: $x \leq y \iff x < y \lor x=y$
$ \textbf{...
- 11.3k
11
votes
2
answers
314
views
Spaces with every compactification $0$-dimensional which aren't locally compact
Recently I've proven the following theorem
Theorem. Let $X$ be a zero-dimensional locally compact Hausdorff space. Then the following are equivalent:
Every compactification of $X$ is zero-dimensional....
- 1,211
6
votes
0
answers
218
views
Hölder's inequality for trace-class maps of $p$-liquid spaces and a related conjecture of Grothendieck
In Condensed Math and Complex Geometry Proposition 8.8, Clausen-Scholze describe trace-class maps between projective objects in the $p$-liquid category as sums of rank 1 operators against ${<}p$-...
- 61
2
votes
0
answers
52
views
On distributions and kernels
Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
- 1,429
3
votes
1
answer
194
views
Opposite convex order on the set of positive roots of a semisimple Lie algebra
Let $\mathfrak{g}$ be a semisimple Lie algebra of rank l and let $\Delta^+$ be its set of positive roots. Denote by $s_1,...,s_l$ the simple generators of its Weyl group and let $w_0$ be the longest ...
1
vote
0
answers
99
views
$L^{\infty}$ estimate for bounded function on complex manifold with conic Kähler metric
Let $\overline{X}$ be a compact Kähler manifold of complex dimention $n$ with normal crossing divisors $D=\sum_{i=1}^{m}=D_{i}$. For $0<\alpha< 2$, we can construct a conic Kähler metric by ...
- 206
3
votes
0
answers
147
views
Tate conjecture for singular varieties in terms of intersection homology
In his book “Mixed motives and algebraic K-theory”, Jannsen generalizes the Tate conjecture to a potentially singular projective variety $X$ over a finitely generated field. The statement is the same ...
- 658
0
votes
0
answers
145
views
Bundles on stacks
We want to define what is a morphism of bundles over an algebraic stack. If $X$ is an algebraic stack and $V_n$ is the stack of rank $n$ vector bundles, a vector bundle on $X$ will be a morphism of ...
- 11
4
votes
1
answer
176
views
Every elliptic surface contains only finitely many negative self-intersection rational curves?
By a properly elliptic surface, I mean an algebraic surface $X$ with Kodaira dimension $\kappa(X)=1$. It has a natural elliptic fibration $\pi\colon X\rightarrow S$.
According to section 5.2 of this ...
- 141
1
vote
0
answers
67
views
Quasi-geodesics in Alexandrov spaces
I am trying to understand the notion of quasi-geodesic in Alexandrov space with curvature bounded below following the Perelman-Petrunin paper. I have two questions:
Is it true that the shortest ...
- 21.8k
4
votes
1
answer
285
views
Known cases of Tate conjecture for varieties which are smooth over a curve
What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve? I am ...
- 658
6
votes
2
answers
324
views
Set theoretical foundations for derived categories
A modern approach to derived functors, that has been shown to be useful in a number of different circunstances is that of a derived category (see the book by Yakutieli, for example, here).
However, it ...
- 3,318
0
votes
0
answers
89
views
Extend algebraic morphism to a compactification with normal crossing boundary
Suppose $X$ and $Y$ are smooth algebraic variety over a char $0$ field $k$, and $f:X\to Y$ a morphism. I want to ask whether there exists compactifications $\bar X$ and $\bar Y$ such that $\bar X\...
- 785
3
votes
1
answer
103
views
References: rigorous algorithms for elementary computations in base-b with complexity estimates
Definitions/Notation: Fix positive integers $b$ and $M$. Consider the set of real numbers which can be exactly expressed with $2M+1$ coefficients in base $b$, defined by
$$\mathcal{X}(b,M):=\{x\in \...
- 5,405
41
votes
3
answers
3k
views
How many natural operations on subsets are there?
How many operations are there on subsets of a set that are compatible with images of maps? Of course, we can take unions, but are these all? Spoiler: No.
To formalize this, let $P : \mathbf{Set} \to \...
- 63.1k
0
votes
1
answer
38
views
Sufficient conditions for a sum of Hadamard products of positive semidefinite matrices to be positive definite
Let $A_i$, $B_i$ be hermitian $n$ by $n$ positive semidefinite matrices of rank $1$, for $i = 1, \dots, n$. Assume that the rank of $A_i \circ B_i$ is also $1$, for $i = 1, \dots, n$, where $\circ$ ...
- 5,215
1
vote
1
answer
67
views
Recursive formula for divided differences
In general, a function $f(\cdot)$ defined at points $x_1,x_2,\dots, x_k$, the $(k − 1)$th-order
divided difference is defined by the recurrence relation:
$$
f[x_1,x_2,\dots...x_k]=\frac{f[x_2,\dots......
- 147
3
votes
1
answer
257
views
Asymptotic parametrization for negatively curved surfaces
Let $S$ be a complete simply connected negatively curved surface immersed in Euclidean space $\textbf{R}^3$. Does there exist a parametrization $f\colon\textbf{R}^2\to\textbf{R}^3$ for $S$ such that ...
- 7,179
0
votes
1
answer
119
views
Nonstationary phase method for oscillatory integral
I want to approximate an integral of the form $$\int_a^bf(t)e^{ig(t)}dt,$$where $f(t)$ is smooth, $g(t)$ is real-valued and smooth.
The stationary phase method says that if $t_0\in [a,b]$ is such that ...
- 107
1
vote
0
answers
78
views
Conjecture about Euler quotients related to non-Wieferich numbers $W(n)=\frac{2^n+1}{3}$
For odd natural $n$ define the Euler quotient:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
$a(n)=0$ is $n$ being Wieferich number (not necessarily prime).
For odd $n$,...
- 25.4k
3
votes
0
answers
136
views
Possible gaps for a function and its Fourier transform
This is another question on the possible shape of sets $A,B\subset \mathbb{R}^d,d\geq 2,$ where resp. a non-null Schwarz function $f$ and its Fourier transform can vanish.
A nice remark by Christian ...
- 1,352
2
votes
1
answer
121
views
Fundamental domain of an involution on a manifold
What is known about the fundamental domain of an involution on a manifold? Eg., if the involution is free is it true that there exists a fundamental domain which is a smooth manifold with boundary?
- 658
1
vote
0
answers
124
views
Relation of automorphic representation and its constant term
Let $\pi$ be an irreducible non-cuspidal automorphic representation of a classical group $G$ defined over a number field $F$. Let $P$ be a maximal parabolic subgroup of $G$. Let $\pi'$ be the ...
- 1,019
3
votes
0
answers
167
views
Are motives of K3 surfaces of abelian type?
I refer to the article of van Geemen https://arxiv.org/pdf/math/9903146. What van Geemen calls the Kuga-Satake-Hodge conjecture suggests that for a K3 surface $X$ over $\mathbb{C}$, the summand $h^2(X)...
- 658